Question

Find the radius of the biggest ball that can fit inside a $5cm$ tall cylinder with a volume of $45$ cubic cm.

Answer 1

Hint: Biggest ball will have diameter equal to the diameter of the cylinder. Hence we need to find the diameter of the cylinder first and then we can directly find the radius.

Complete step-by-step answer:
Finding the diameter of the cylinder.
Given,
Height$=5cm$
Volume$=45c{{m}^{3}}$
As we know the formulae for the volume of the cylinder is $\pi {{r}^{2}}h$
Where r is the radius of the cylinder and h is the height of the cylinder.
As we are provided with the height of the cylinder and the volume of the cylinder we can find the radius of the cylinder by putting value in that.
$\begin{align}
  & \pi {{r}^{2}}h=45 \\
 & \pi {{r}^{2}}5=45 \\
 & \pi {{r}^{2}}=9 \\
 & r=\dfrac{3}{\sqrt{\pi }} \\
\end{align}$
Now, as we got the radius of the cylinder.
The radius of the largest ball that can fit inside the cylinder will also be same. Hence, no further calculations are required.

Note: As we have assumed here that the largest ball that can be fitted will have equal diameter as that of the cylinder. We can look at this fact in some other way like we start from the center of the cylinder and start increasing in 360 degree then all we can reach at max is the boundaries of the cylinder and hence we assume that the diameter of the largest ball will equal to the diameter of the given cylinder.
We can also approach this question using the maxima and minima method.